Question

What is the square root of $144-{{x}^{2}}$?

Answer 1

Hint: From the given question we are asked to find the square root of the expression $144-{{x}^{2}}$. For solving this question we will use the definition of square root which is a square root of any number is a number which, if multiplied by itself, produces an original number. Using this and we simplify the expression using the basic mathematical operations and etc…, we will solve this question.

Complete step by step solution:
Let us begin by defining the square root of a number. It is a value, which on multiplication by itself gives us the original number.
When we extend this to the case of an algebraic expression containing both variables and constants, this might or might not hold true. So, if we consider $2+x$, then we cannot simplify further and get its square root. If we consider ${{x}^{2}}+4x+4$, then we know that we can express it as ${{(2+x)}^{2}}$. This in turn tells us that the square root of ${{x}^{2}}+4x+4$ is $2+x$.
Now let us come to this problem. We have the expression as $144-{{x}^{2}}$ . If we try to factorise it, using the fact that $114={{12}^{2}}$ and ${{x}^{2}} = x$, then we would get a form of ${a}^{2}-{b}^{2}$. We can further write it as $(12-x)(12+x)$.
Now, from the above, it is clear that since we do not get the same factors, we cannot have a square root of the given algebraic expression of a square root rather than $\sqrt{144-x^{2}}$.
$\Rightarrow \sqrt{144-{{x}^{2}}}=\sqrt{\left( 12-x \right)\left( 12+x \right)}$

Note: Students must be very careful in doing the calculations. Students should have good knowledge in the square root and its conditions. We should know that for a square root to exist for the $144-{{x}^{2}}$ the x must lie in the range of $-12\le x\le 12$ or else it cannot have a square root among the real numbers.